An efficient algorithm for the solution of min-max problems in multiaxial fatigue

Numerous fatigue criteria require the evaluation of critical quantities to estimate fatigue damage. The approach based on the construction of the minimum circle or hypersphere is often employed in multiaxial high-cycle fatigue criteria to compute critical quantities as the amplitude and mean value of the shear stress. Such a construction can be mathematically formulated as a min-max optimization problem for which efficient numerical strategies are of utmost importance. The present paper proposes a novel algorithm for solving such an optimization problem. The algorithm is based on the alternating direction method of multipliers (ADMM) and is very simple to implement. A wide range of fatigue loading paths, formulated in the two-dimensional and deviatoric spaces, are considered for numerical testing. Comparisons with reference solutions and with two alternative optimization approaches demonstrate the accuracy as well as the high efficiency of the proposed ADMM-based algorithm. A discussion on the adopted stopping criterion and on algorithm parameters is also addressed.